Anti-deadzone adaptive fuzzy dynamic surface control for planar space robot with elastic base and flexible links

In order to combat the impact of the dead zone and reduce vibration of the space robot's elastic base and flexible links, the trajectory tracking and vibration suppression of a multi-flexible-link free-floating space robot system are addressed. First, the elastic connection between the base and the link is considered as a linear spring. Then the assumed mode approach is used to derive the dynamic model of the flexible system. Secondly, a slow subsystem characterizing the rigid motion and a fast subsystem relating to vibration of the elastic base and multiple flexible links are generated utilizing two-time scale hypotheses of singular perturbation. For the slow subsystem with a dead zone in joint input torque, a dynamic surface control method with adaptive fuzzy approximator is designed. Dynamic surface control scheme is adopted to avoid calculation expansion and to simplify calculation. The fuzzy logic function is applied to approximate uncertain terms of the dynamic equation including the dead zone errors. For the fast subsystem, an optimal linear quadratic regulator controller is used to suppress the vibration of the multiple flexible links and elastic base, ensuring the stability and tracking accuracy of the system. Lastly, the simulation results verify the effectiveness of the proposed control strategy.

1. Based on two-time scale assumptions, the system is divided into a slow subsystem representing the rigid motion and a fast subsystem describing the flexible vibrations.For the slow subsystem, a dynamic surface controller with an adaptive fuzzy approximator is designed to tackle dead zone.The application of dynamic surface avoids the calculation expansion caused by backstepping method and reduces the calculation amount.2. The fuzzy logic function approximates the dynamic uncertainty including dead zone error, so that the desired point to point trajectory tracking of the base's attitude and joints angle can be achieved.3.For the fast subsystem, the linear quadratic regulator (LQR) is used to suppress the flexible vibration of the base and links concurrently.
There are six sections in this essay.In section "System dynamics equation", the dynamic equations for a planar space robot with an elastic base and multiple flexible links are determined.In Section "Decomposition of fast and slow systems", based on the singular perturbation method, the system is decomposed into the fast and slow subsystems.Section "Design of combined control law" designs an anti-deadzone adaptive fuzzy dynamic surface control method for the slow subsystem and a linear quadratic controller for the fast subsystem.Section "Simulation experiment" shows the simulation results and analysis comparison of a space robot with an elastic base and two flexible links.Section "Conclusions" gives the conclusion.

System dynamics equation
Taking the space robot with elastic base and multi-flexible links as the research object, the system consists of a free-floating base B 0 and multi-flexible links B i (i = 1, 2 • • • , n) .The model is shown in Fig. 1.The elasticity of the guide rail is simplified as a light spring to represent the elasticity of the base, and the elastic displacement is denoted as χ .It is assumed that: (1) the spring is a massless spring; (2) The spring only performs the pulling and retracting movement along the axis; (3) Spring elasticity coefficient k χ is constant; (4) The initial displacement of the spring is zero.Establish the conjoined coordinate O i x i y i of each split B i (i = 0, 1, . . ., n) , where O 0 coincides with the centroid O c0 of B 0 , and O i (i = 1, 2, . . ., n) is the center of the corresponding rotary hinge.The light spring connects O 1 and B 0 , the x i (i = 0, . . ., n − 1) axis is collinear with O i O i+1 (i = 0, . . ., n − 1) , and the x n axis and B n are always tangent to O n .In the initial state, the distance between O 1 and O 0 is l 0 .The length of B i (i = 1, 2, . . ., n) along the x i axis is l i .C is the total centre of mass of the system.The mass of the base and the moment of inertia about the centre of mass are m 0 and J 0 , respectively.The translational inertial coordinate system (O − XY ) is established with any point O in space as the origin.
The links are assumed to be slender and homogeneous, and vibrate transversely in the plane.The bending deformations are mainly considered, and the axial and shear deformations are neglected.The linear density of flexible arm and the bending stiffness of the section is (EI) i .According to the theory of vibration mechanics, the flexible links can be thought of as Euler Bernoulli beams, and their elastic deformations can be recorded as: where σ i (X i , t) is the transverse elastic deformation of B i at section X i (0 ≤ X i ≤ l i ) , ω ij (X i ) is the modal function of the j th order of B i , ξ ij (t) is the modal coordinate corresponding to ω ij (X i ) , and κ i is the number of truncated terms.In this paper, simply supported beams B i (i = 1, 2, . . ., n − 1) and cantilever beams B n , respectively, are considered.
As shown in Fig. 1, it can be seen that the vector r δi (i = 1, 2, . . ., n) of any point on each flexible link relatives to the origin O of the inertial coordinate system (O − XY) can be expressed as: where e xj = cos j k=0 q k sin j k=0 q k T , and e yj = − sin Then according to the total centre of mass theorem of the system where m 0 + n i=1 ρ i l i = M r 0 and r δi (i = 1, 2, . . ., n) are deduced as (1) where L j (j = 0, 1, . . ., 2n) are the combined functions of the inertia parameters of the system.Take the derivative of Eq. ( 5) to time t , the velocity vector ṙδi (i = 1, 2, . . ., n) can be obtained.The total kinetic energy T of the space robot system with elastic base and flexible links can be expressed as: where T 0 = 1 2 m 0 ṙ2 0 + J 0 q2 0 , and T δi = 1 2 ρ i l i 0 ṙ2 δi dX i (i = 1, 2, . . ., n) Because the space robot system is in weightlessness in the outer space environment, the total potential energy U of the space robot system with elastic base and flexible links is derived from the sum of the bending strain energy of the flexible links and the elastic potential energy of the elastic base: Without losing of generality, assuming that the initial momentum of the system is zero, that is, ṙc = 0 .The dynamic equations of the fully elastic space robot with elastic base and flexible links can be obtained by using the Lagrange equation of the second kind: T is the rigid generalized coordinate column vector of the base's attitude and the relative rotation angle of the joints of the links,ξ = ξ 11 , . . ., ξ 1κ 1 , . . ., ξ n1 , . . ., ξ nκ n T is the generalized coordinate column vector of the flexible modes of the links, is a symmetric and positive definite mass matrix.hq, δ, q, δ qT δT 0 ω ′′T ij ω ′′ dx i is the stiffness matrix of the links.τ ∈ R n+1 is the torque column vector of the base and links' joints.

Decomposition of fast and slow systems
During the operation of the space robot, with the movements of links and base, not only do the multiple flexible links deform and vibrate, but the base also oscillates.The dynamic model is divided into a slow subsystem of rigid motion and a fast subsystem that reflects base elasticity and flexible vibration of links in order to accomplish high-precision control and vibration suppression of space robots.As a result, Eq. ( 5) is written as a block matrix: .
. Because M is a symmetric and positive definite matrix, its inverse exists: Defining singular perturbation scale factors ε 2 = 1/ min k 11 , . . ., k 1κ 1 , . . ., k n1 , . . ., k nκ n , k χ ,and new vari- able z , K zε 2 = δ, K = ε 2 K .The singular perturbation model of the space robot system with elastic base and flexible links can be obtained from Eq. ( 9): According to the singular perturbation model of the system in Eqs. ( 11) and ( 12), the following combined control law is designed: (5) where τ s is the control torque of the slow subsystem to realize the angle tracking of the base and joints, and τ f is the control torque of the fast subsystem to suppress the vibrations caused by the elastic base and multi-flexiblelinks at the same time.
In order to deduce the slow subsystem of the space robot with elastic base and flexible links, ε is set to zero firstly.Then the slowly varying manifold expression z of the system can be solved from Eq. ( 12): where the matrix or variable with the dash "￣" means the corresponding slowly varying component.
Substituting the above equation into Eq.( 13) and considering ff N fs , the slow subsystem is obtained: In order to obtain the fast subsystem, defining new variables, and making p 1 = z − z , p 2 = ε ż , Eq. ( 12) can be rewritten as: Adding the fast variable time scale ̟ = t/ ε , let ε = 0 .Then the dynamic equation of fast subsystem is: It describes the vibration of the elastic base and flexible links.

Design of Anti-deadzone adaptive fuzzy dynamic surface controller for slow subsystem
The base position and attitude of the space robot are usually adjusted by momentum wheels or reaction jet devices.The joint hinges are driven by motors, so there is a dead zone in joint input torque.
Considering dead zone in joint input torque, the slow subsystem can be written as: where τ s = τ 0 D T (τ r ) T ∈ R n+1 is the actuator output torque τ 0 of the base and the joints' actuator output T of the links in the slow subsystem, and D(τ r ) is the column vector with the dead zone of the joint input torque τ r = (τ 1 , . . ., τ n ) T .
Since it is typically challenging to acquire the exact parameters of the system dynamics model, let υ 1 = q , υ1 = υ 2 = q .Then, the slow subsystem can be written as state equation in the following form: w h e r e Mss , Ĥss a r e t h e n o m i n a l m o d e l s , Dead zone in joint input torque "Dead zone" refers to the range where change in input has no effect on output.It reflects the input-output relationship of zero output after the input of joint torque enters the dead zone.When the signal enters the dead zone, there will be a certain loss, which will result in the deviation of system control.
The joint input torque is τ r and the joints' actuator output torque is D(τ r ) .The simplified dead zone model can be expressed as 36 : where b li < 0 and b ri > 0 represent the dead zone's left and right relative widths.m li and m ri represent the left and right slopes of the dead zone, respectively, m ri and m li are specified to be greater than zero.
The difference between input and output of dead zone is expressed as: www.nature.com/scientificreports/Therefore, Eq. ( 20) is rewritten as follows: where τ c = τ 0 τ T r T is the torque column vector of the base and joints before passing the "dead zone".

Controller design
In this study, the dynamic surface control technique is used to create the virtual control variable and control input signal for the space robot's slow component.This information serves as the foundation for the construction of the slow subsystem's control law τ c , which enables actual trajectory q of the space robot's rigid motion to follow the anticipated trajectory q d despite having an elastic base and flexible links.
The design steps of dynamic face control are as follows: Step 1: Define the first error surface as: Design a virtual control variable υ2 : where design constant c 1 >0.With υ2 as the input and υ 2d as the new state variable output, a first-order low-pass filter (LPF) is introduced: where the time η 2 > 0.
Step 2: Define the second dynamic surface in order to create the control law of the slow subsystem: Substituting Eq. ( 23) into the first derivative of Eq. ( 27), we can obtain: In the study, the fuzzy logic system is used to approximate (A) = 0 −D T � T + T � , whose exact value is unknown.
(A|W ) is utilized to approximate (A) , then where W is the weight matrix, O(A) = [O 1 , . . ., O γ ] T is the fuzzy basis vector, and γ is the number of rules.
is the fuzzy basis network input.The fuzzy basis function O k is expressed as: Select the Gaussian membership function: where a jk and b jk represent the Gaussian membership function's center and breadth, respectively.The optimal value W * of W is a constant matrix and satisfies where W * is bounded, that is, there is a normal number ρ W , which satisfies �W� ≤ ρ W . Then (A) is expressed as follows: where µ * is the approximation error.
The slow subsystem's control rule is made to: where Ŵ and μ are the estimated values of W * and µ * respectively, and c 2 > 0.

Linear quadratic controller for fast subsystem
In this work, the linear quadratic regulator is used to control the fast subsystem of the space robot with elastic base and flexible links, so as to actively suppress the vibration of the elastic base and flexible links at the same time.
Writing the fast subsystem Eqs. ( 17) and ( 18) into the state equation expression, and making the state variable P = p 1 p 2 T , the Eqs.( 17) and ( 18) are combined as follows: where Equation (55) demonstrates that the fast subsystem is a linear system, and that the system state variable P can be adjusted to zero by using the optimal control method, thereby achieving the suppression of base elasticity and flexible links vibration.For a linear system, if the performance index function is defined as the integral of the quadratic function with respect to the state variable and the control variable, the system can obtain the optimal performance by finding the control τ f when the function takes the minimum value.
The function for the linear quadratic optimal control performance index indicator is presented as follows: where is a symmetric weighted matrix with a positive semi-definite, and n+1) is a symmetric weighted matrix with a positive definite.
(48) www.nature.com/scientificreports/According to the linear quadratic optimal control theory, in order to minimize ϒ , the control quantity should be created as follows: where fulfills the following Riccati algebraic equation.
Figure 2 shows the block diagram of the control scheme.

Simulation experiment
Taking the space robot model system with elastic base and two flexible links shown in Fig. 3 as an example, the numerical simulation experiment is carried out, using a Lenovo Thinkpad X1 with an Intel Core i7-10510U processor, 16 GB memory, Windows 11 and MATLAB.The values of parameters for space robot system and controller are shown in Table 1.The dynamic surface control with adaptive fuzzy approximator (DSC_wAFA) is compared with dynamic surface control without adaptive fuzzy approximator (DSC_woAFA) to evaluate performance of the proposed method.
Figures 4, 5 and 6 shows the trajectory tracking of rigid motion of the base 's attitude and two joint angles of the space robot system using DSC-wAFA and DSC-woAFA in the case of dead zone in joint input torque.When the results of the DSC-wAFA is compared with and DSC-woAFA, it is shown that the actual trajectory of the base 's attitude and two joint angles can track the desired trajectory effectively with proposed control method and there is obvious tracking error of two joint angles in the DSC-woAFA system.Figures 7 and 8 show the trajectory tracking errors of the base 's attitude and two joint angles utilizing DSC-wAFA and DSC-woAFA, respectively.It can be seen from the partial enlarged view of Fig. 7 that, after about 18s , the steady-state errors of base's attitude, joint angle 1 and joint angle 2 are within 1.1 × 10 −3 rad , 5 × 10 −4 rad and 1.1 × 10 −4 rad respectively.Therefore, the proposed anti-deadzone controller gives satisfactory performance.Through comparison, it can be seen that when the dead zone adaptive fuzzy approximator is turned off, the tracking error of base's attitude, joint angle 1 and joint angle 2 are within 1.4 × 10 −3 rad , 0.03 rad and 0.1 rad respectively, and the tracking error of both joints cannot converge due to damage of the deadzone in joint input torque of two links.
The length of B i (i = 1, 2) along the x i axis l 1 = 1.5 m , l 2 = 1.0 m The mass of the base m 0 = 40 kg The moment of inertia about the centre of mass J 0 = 34.17kg m 2 The linear density of flexible arm B i (i = 1, 2) The bending stiffness of the section (EI) 1 = 50 N m 2 , (EI) 2 = 50 N m 2 Spring stiffness coefficient The number of fuzzy rules γ = 5 Slow change subsystem regulation parameters: The expected configuration of the attitude angle terminal of the base and the two joints The initial configuration The initial displacement of the base spring χ(0) = 0 The number of truncated terms Set the track tracking process simulation time t = 20s      shows the flexible modes of two links of the space robot system using DSC-wAFA and DSC-woAFA in the case of dead zone.The first mode of B 1 attenuates from 0.11m at 1.5s to zero in the DSC-wAFA system, but vibrates between ±0.02m after 4s in the DSC-woAFA system.The second mode of B 1 attenuates from 0.005m at 0.2s to zero in the DSC-wAFA system, but vibrates between ±0.012m after 4s in the DSC-woAFA system.The first mode of B 2 attenuates from 0.027m at 0.7s to zero in the DSC-wAFA system, but vibrates between ±0.005m after 4s in the DSC-woAFA system.The second mode of B 2 attenuates from 0.008m at 0.06s to zero in the DSC-wAFA system, but vibrates between ±0.0005m after 4s in the DSC-woAFA system.Figure 13 shows the elastic vibration of the base of the space robot system.The elastic displacement is 0.011m at 1.58s and attenuates to zero after 15s in the DSC-wAFA system, but vibrates between ±0.0015m after 4s in the DSC-woAFA system.It is demonstrated that, when the rigid motion trajectory is stable in the DSC-wAFA system, the elastic oscillation of the base and the vibration of the two flexible links are suppressed by using linear quadratic controller.Thus, the effectiveness of the vibration suppression scheme is verified.
Figures 14 and 15 show the control torques of the base 's actuator and two joints utilizing DSC-wAFA and DSC-woAFA respectively.The control torques converge to zero when the DSC-wAFA system is stable, but are oscillating in the DSC-woAFA system.
From the simulation results of the proposed controller, it can be seen that the base's attitude and the two joint angles can track the desired trajectory of rigid motion, and the vibration of elastic base and flexible links are damped out at the same time.Therefore, the proposed control scheme can tackle the effect of dead-zone and dual vibration effectively, with high tracking accuracy and satisfactory performance.

Conclusions
1. Considering the multiple coupling between the elastic base and the flexible links, the dynamic equations of the space robot with elastic base and flexible links are derived by integrating the momentum conservation relation of the system, the second kind of Lagrange equation and the assumed mode method.2. Based on the singular perturbation method, the system is decomposed into slow and fast subsystems, which describe the rigid motion, the base elasticity, and the flexible vibrations of links respectively.For the slow subsystem, a dynamic surface controller with adaptive fuzzy approximator is designed when there is dead zone in joint input torque.For the fast subsystem, optimal quadratic controller is adopted.The combined control scheme can not only overcome the adverse effects of the dead zone on the system, ensure that the space robot system can track the desired trajectory of the rigid motion, but also actively suppress the vibration of the base elasticity and the flexible links at the same time, overcome the impact of unknown inertia parameters, and meet the control requirements.

Figure 2 .
Figure 2. The adaptive fuzzy dynamic surface control scheme for space robot.

2 χFigure 3 .
Figure 3. Space Robot System with Elastic base and Flexible Links.

Figure 10 .Figure 11 .
Figure 10.The second mode of flexible link B 1 .

Figures 9 ,
Figures 9,10, 11 and 12  shows the flexible modes of two links of the space robot system using DSC-wAFA and DSC-woAFA in the case of dead zone.The first mode of B 1 attenuates from 0.11m at 1.5s to zero in the DSC-wAFA system, but vibrates between ±0.02m after 4s in the DSC-woAFA system.The second mode of B 1 attenuates from 0.005m at 0.2s to zero in the DSC-wAFA system, but vibrates between ±0.012m after 4s in the DSC-woAFA system.The first mode of B 2 attenuates from 0.027m at 0.7s to zero in the DSC-wAFA system, but vibrates between ±0.005m after 4s in the DSC-woAFA system.The second mode of B 2 attenuates from 0.008m at 0.06s to zero in the DSC-wAFA system, but vibrates between ±0.0005m after 4s in the DSC-woAFA system.Figure13shows the elastic vibration of the base of the space robot system.The elastic displacement is 0.011m at 1.58s and attenuates to zero after 15s in the DSC-wAFA system, but vibrates between ±0.0015m after 4s in the DSC-woAFA system.It is demonstrated that, when the rigid motion trajectory is stable in the DSC-wAFA system, the elastic oscillation of the base and the vibration of the two flexible links are suppressed by using linear quadratic controller.Thus, the effectiveness of the vibration suppression scheme is verified.Figures14 and 15show the control torques of the base 's actuator and two joints utilizing DSC-wAFA and DSC-woAFA respectively.The control torques converge to zero when the DSC-wAFA system is stable, but are oscillating in the DSC-woAFA system.From the simulation results of the proposed controller, it can be seen that the base's attitude and the two joint angles can track the desired trajectory of rigid motion, and the vibration of elastic base and flexible links are damped out at the same time.Therefore, the proposed control scheme can tackle the effect of dead-zone and dual vibration effectively, with high tracking accuracy and satisfactory performance.

Table 1 .
The values of parameters.